Question

Question 4 If x – iy = [(a-ib)/(c-id)]^(1/2) prove that (x^2 + y^2)^2 = (a^2 + b^2)/(c^2 + d^2)

Class X1 - Maths -Complex Numbers and Quadratic Equations Page 112

Answer 1

x - iy = √{(a -ib)/(c-id)}

concept :-
|z|ⁿ = |zⁿ| and also |z1/z2| = |z1|/|z2|

solution:-
x -iy = √{(a-ib)/(c-id)}
x + (-y)i = √{(a+ (-b)i)/(c+(-d)i)}
now, take modulus both sides,

|x + (-y)i| = |(a+(-b)i)/(c+(-d)i)|½
√(x² + y²) = |(a+(-ib))|½/|(c+(-d)i)|½

take square both sides,
(x² + y²) = |a +(-b)i|/|c+(-d)i|
(x²+y²) = √(a²+b²)/√(c²+d²) [ by using formula |(±a)+i(±b)| = √(a²+b²) ]
Take square both sides,

Hence,
(x² + y²)² = (a²+b²)/(c²+d²)